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I know what universal and existential quantifiers are but following is confusing,may be its comibination of set notation and quantifers. What does the following statement means?

∀xP(x)
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2 Answers 2

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For all $x$, the statement $P(x)$ is true.

It helps to separate things out: $\forall x$, $P(x)$.

Edit to answer your question: $P(x)$ is just a statement. It is just some property of the $x$s. Something we say about $x$. For example, $P(x)$ could be $x>0$ or "$x$ is blue", or $x \in \mathbb{Z}$.

So $P(x)$ is not a set, is is just a statement about the $x$s. However, we can use $P(x)$ to make a set. Suppose $P(x)$ means $x>0$. Then we could have $$A = \{x \in \mathbb{R} \mid x > 0\}$$ Which reads "A is the set of all real numbers $x$ such that $x$ is greater than zero."

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@Labeeb edited my post: hope that helps –  Newb Nov 29 '13 at 15:08

Here, $P(x)$ denotes a statement we call a predicate which applies to $x$, asserting, in effect, that "$x$ is $P$."

$P(x)$ might mean, for example, "$x$ is purple", or else maybe, $P(x)$ denotes "$x$ is an element of the set $P$."

$\forall x$ denotes that we are asserting something that holds for all $x$ in a specified domain.

$\forall x\; P(x)\; $ then asserts that for all $x$ in the domain, $\,P(x)\,$ is true. For example, if $P(x)$ means "x is purple", then $\,\forall x\; P(x)\,$ would mean: "All $x$ in the domain are purple." If $P(x)$ denotes "$x$ is an element in the set $P$," then $\forall x\,P(x)$ would denote "For all $x$ in the domain, $x \in P$."

Important

For $\forall x\,P(x)$ to be meaningful, we need to have some sort of "translation key" telling us what the relevant domain is, and telling us what the predicate $P(x)$ denotes. Otherwise, all we can say is

"For some specified domain and some predicate $P(x)$, for every member $x$ in the domain, it is true that $P(x)$."

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Needs another UV +1 –  Amzoti Nov 30 '13 at 1:15

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